Necessary and Sufficient Conditions for the Solvability of the Lp Dirichlet Problem on Lipschitz Domains

By Zhongwei Shen. 2005

59 citations recorded by the source corpus.

We study the homogeneous elliptic systems of order $$2\ell$$ with real constant coefficients on Lipschitz domains in$$\mathbb{R}^n$$, $$n\ge 4$$. For any fixed p > 2, we show that a reverse Hölder condition with exponent p is necessary and sufficient for the solvability of the Dirichlet problem with boundary data in Lp. We also obtain a simple sufficient condition. As a consequence, we establish the solvability of the Lp Dirichlet problem for $$n\ge 4$$ and $$2-\epsilon< p<\frac{2(n-1)}{n-3} +\epsilon$$. The range of p is known to be sharp if $$\ell\ge 2$$ and $$4\le n\le 2\ell + 1$$. For the polyharmonic equation, the sharp range of p is also found in the case n = 6, 7 if $$\ell=2$$, and $$n=2\ell+2$$ if $$\ell\ge 3$$.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/0b58a978a6f52b07cf41f873012e5efb36ab40a8

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Homogenization of elliptic boundary value problems in Lipschitz domains (2009, methodology): “Our approach to the range 2 < p < 2 + δ, which is based on a real variable argument developed by Shen [25, 26], is different from that used in [13, 21].”
The dirichlet problem in lipschitz domains for higher order elliptic systems with rough coefficients (2007, background): “when p is near 2, i.e., 2 − e 0 depending on the Lipschitz character of . On p.2 of [49] the authors ask whether the Lp-Dirichlet problem for these operators is solvable in a given Lipschitz domain for p ∈ (2 − e, 2(n−1) n−3 + e), and a positive answer has been recently given by Z. Sh en in [51].”
Layer potential methods for elliptic homogenization problems (2009, methodology): “Using the L techniques developed in [9, 24, 25] for constant coefficients, we may further extend the results for the Dirichlet problem with L boundary data to the range 2 < p ≤ ∞ if d = 3, and to 2 < p < 2(d−1) d−3 + δ if d ≥ 4.”